1. Mathematical Model of Motion Chapter 5

2. Velocity Equations Average velocity: v =  d/  t To find the distance traveled with constant or average velocity : d = vt To find an object’s location along the x axis: x = x 0 + vt

3. Constant Acceleration Equations Final velocity after acceleration: v = v 0 + at Final position after acceleration  d = ½ (v + v 0 )t  d = v 0 t + ½at 2 Final velocity after acceleration v 2 = v 0 2 + 2a(  d)

4. Free Fall Objects accelerate as they fall Acceleration due to gravity is 9.80 meters per second per second g = 9.80 m/s 2 So speed of falling object increases almost 10 m/s every second it falls

5. Free Fall Calculations Instantaneous speed of falling object = gt acceleration times the elapsed time of fall To find distance of fall, d = 1/2 gt 2 In any time interval, distance fallen = avg. speed times elapsed time

6. Terminal Velocity Air resistance slows objects, depending on surface area and air density Terminal velocity (speed) is reached when force of air resistance equals force due to gravity In vacuum, all objects fall at same rate

7. Graphing Motion: position vs. time Plot position (dependent variable) versus time (independent variable) Sometimes d is used to represent position In horizontal motion, x is often used instead To find position at any time, read the values off the graph using the line

8. Graphing Motion: Position Graph with 2 or more objects Motion for two or more objects can be plotted on the same graph Where lines cross, objects are at the same position

9. Graphing Motion: Position Graph with Uniform Motion Equal displacements in equal time intervals Produces linear graph Slope of position graph gives velocity Horizontal line means zero velocity Upward slope means positive velocity Downward slope means negative velocity

10. Position Graph with Uniform Motion Motion diagram to graph

11. Slope of a Graph

12. Graphing Motion: Position Graph with Acceleration Graph will be curved Constant acceleration gives parabola Slope of graph at any point gives instantaneous velocity

13. Position Graph with Acceleration Converting motion diagram to graph

14. Velocity vs Time Graphs Plot velocity as dependent variable, time as independent variable Constant velocity gives horizontal line Slope of graph gives acceleration Positive acceleration gives upward slope Negative acceleration slopes downward If graph is linear, acceleration is constant

15. Velocity vs Time: acceleration

16. Comparing Uniform and Accelerated Motion Graphs Uniform motion Accelerated Motion

17. Comparing Positive and Negative Velocity

18. Speeding up and Slowing Down

19. Velocity vs Time Graphs: Finding Displacement Displacement can be found from velocity graph by finding the area between the graph and the time axis Divide the area bounded by the graph line, the horizontal axis and the initial and final times into geometric sections (squares, rectangles, triangles) and find the area Area below the time axis is negative displacement

20. Area under the Velocity Graph

21. Area Under the Velocity Graph Divide complex areas into triangles and rectangles